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\title{\vskip -70pt           % move title up
\huge The Ellipsoid 
\footnote{This file is from the 3D-XploreMath project. You can find it on the web by searching the name.}
\vskip -20pt  %reduce space between ttle and authorname.
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\author{H.K.}

\begin{document}

\maketitle

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\LARGE


\cl{ See in Documentation: About Quadratic Surfaces.}
\LF
\centerline{\includegraphics{Ellipsoid.png}}
\LF
The Ellipsoid in 3D-XplorMath is parametrized as
$$
  x = aa\cdot \cos u\cdot \sin v, \hskip3mm y= bb\cdot  \sin u\cdot \sin v, 
  \hskip3mm z = cc\cdot  \cos v,
$$
with the default $aa= bb=1.5,\  cc=2.0$.
\Lf
In 3D-XplorMath the Ellipsoid is shown together with a few rays which leave
one focal point, are reflected in the surface and come together again in the
other focal point. This illustrates the use of the Ellipsoid as {\it Whispering
Gallery}. A whispering gallery may be realized by an Ellipsoid ceiling in a
pub so that the conversations at one table can be heard at another table. 
Whispering galleries were also built in royal parks with some ellipsoid
reflector near a table for visitors placed at one focal point and a hidden chair
for the listener at the other focal point.
\Lf
The first {\tt default Morph} varies the size of the Ellipsoid. One can also select
in the View Menu {\tt Morph Light Source Of Rays} to illustrate that the rays do
not come together at one point unless they start from a focal point.
\Lf
By selecting {\tt Remove Focal Rays} in the Action Menu one returns to the
standard rendering of surfaces. One may turn on  the focal rays only if
{\tt Wireframe Display} is selected.
\Lf
For geometric arguments concerning the focal points see: {\tt Ellipse} in the
Plane Curve category.


\bye